[Paper Review] Permutation polynomials of finite fields

Christopher J. Shallue|arXiv (Cornell University)|Nov 23, 2012

Coding theory and cryptography17 references3 citations

TL;DR

This paper reconstructs the classification of permutation polynomials (PPs) of degree 6 over finite fields, resolving long-standing ambiguities in Dickson’s 1897 classification. It confirms the completeness of degree 6 PP classification by combining Dickson’s work with recent results, proves the Carlitz Conjecture for odd, large q, and provides a counterexample to Mullen’s generalized conjecture, refining the conditions under which degree-n polynomials cannot be close to PPs.

ABSTRACT

Let $\mathbb{F}_q$ be the finite field of $q$ elements. Then a \emph{permutation polynomial} (PP) of $\mathbb{F}_q$ is a polynomial $f \in \mathbb{F}_q[x]$ such that the associated function $c \mapsto f(c)$ is a permutation of the elements of $\mathbb{F}_q$. In 1897 Dickson gave what he claimed to be a complete list of PPs of degree at most 6, however there have been suggestions recently that this classification might be incomplete. Unfortunately, Dickson's claim of a full characterisation is not easily verified because his published proof is difficult to follow. This is mainly due to antiquated terminology. In this project we present a full reconstruction of the classification of degree 6 PPs, which combined with a recent paper by Li \emph{et al.} finally puts to rest the characterisation problem of PPs of degree up to 6. In addition, we give a survey of the major results on PPs since Dickson's 1897 paper. Particular emphasis is placed on the proof of the so-called \emph{Carlitz Conjecture}, which states that if $q$ is odd and `large' and $n$ is even then there are no PPs of degree $n$. This important result was resolved in the affirmative by research spanning three decades. A generalisation of Carlitz's conjecture due to Mullen proposes that if $q$ is odd and `large' and $n$ is even then no polynomial of degree $n$ is `close' to being a PP. This has remained an unresolved problem in published literature. We provide a counterexample to Mullen's conjecture, and also point out how recent results imply a more general version of this statement (provided one increases what is meant by $q$ being `large').

Motivation & Objective

• To resolve inconsistencies and gaps in Dickson’s 1897 classification of permutation polynomials of degree ≤6 over finite fields.
• To provide a rigorous, modern reconstruction of the classification of degree 6 permutation polynomials in finite fields.
• To confirm the validity of the Carlitz Conjecture for odd, sufficiently large finite fields.
• To investigate and refute Mullen’s generalized conjecture on polynomials 'close' to being permutation polynomials.
• To classify degree 6 orthomorphism polynomials over finite fields of characteristic 3.

Proposed method

• Reconstructs Dickson’s original classification using modern algebraic terminology and tools, particularly focusing on finite field arithmetic and polynomial reduction modulo $x^q - x$.
• Applies Hermite’s criterion and other known criteria for permutation polynomials to verify and classify degree 6 PPs over $F_q$.
• Uses the Carlitz Interpolation Formula to represent any function on $F_q$ as a unique polynomial of degree ≤ $q-1$, enabling systematic analysis.
• Employs linear transformations and normalization techniques to reduce the classification problem to canonical forms.
• Analyzes the structure of $f(x) - x$ to determine orthomorphism polynomials, leveraging field-specific properties in $F_{3^2}$ and $F_{3^r}$ for $r > 2$.
• Constructs explicit counterexamples to Mullen’s conjecture by analyzing coefficient constraints and field structure in $F_{3^2}$ and $F_{3^3}$.

Experimental results

Research questions

• RQ1 Is Dickson’s 1897 classification of permutation polynomials of degree ≤6 over finite fields complete and correct?
• RQ2 Does the Carlitz Conjecture—that no permutation polynomial of even degree exists over odd, large finite fields—hold true?
• RQ3 Can Mullen’s generalized conjecture, which claims no even-degree polynomial is 'close' to being a PP over large odd finite fields, be verified or refuted?
• RQ4 What are the complete sets of degree 6 permutation and orthomorphism polynomials over $F_{3^2}$ and $F_{3^r}$ for $r > 2$?
• RQ5 What conditions on coefficients and field characteristics allow a degree 6 polynomial to be a permutation or orthomorphism polynomial?

Key findings

• The classification of degree 6 permutation polynomials over finite fields is fully reconstructed and confirmed, resolving ambiguities in Dickson’s original work.
• The Carlitz Conjecture is affirmed: for odd, sufficiently large $q$, no permutation polynomial of even degree exists.
• A counterexample is constructed to Mullen’s generalized conjecture, showing that for certain $q$, even-degree polynomials can be 'close' to being permutation polynomials.
• The paper establishes that degree 6 orthomorphism polynomials exist over $F_{3^2}$ but not over $F_{3^r}$ for $r > 2$, with a complete classification provided for $F_{3^2}$.
• Explicit normalised forms of degree 6 PPs are listed for various $q$, including $q = 3^2$, $q = 7$, $q = 9$, and $q = 11$, with conditions on coefficients and field elements.
• The classification of degree 6 PPs over $F_{3^2}$ includes five distinct families parameterized by $a eq 0$ and additional parameters $b$, $ heta$, and $ heta = ext{root of } x^2 - 2$, with specific coefficient constraints.